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When I read descriptions of Apollonius' treatise on conics, some of them say that he invented co-ordinate geometry, some say that he kind of did and others are silent on the matter. Or is it the case that there is no simple answer?

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What do you want to know that isn't covered by the Wikipedia article and accompanying references? – Jonas Meyer Mar 22 '10 at 1:19
I'm not a historian of mathematics, but a reasonable-seeming starting point would be to see what Morris Kline has to say in and then to go from there to see if new readings/findings have emerged. (Questions of invention are tricky in history of ideas/science, since it may not be the case that the "inventors" think along the same lines we do. E.g. who "invented" integration?) – Yemon Choi Mar 22 '10 at 1:41
I think questions of history can be extremely appropriate for this site. This question, however, could be improved. For example, you could have included a short bibliography: which descriptions of Apollonius have you read? – Theo Johnson-Freyd Mar 22 '10 at 2:42
Conc. "descriptions of Apollonius' treatise on conics" : Why not the treatise itself? Only that can give an appropriate impression. As far as I remember my one's: He used problem specific coordinates, but had not sized the idea down to the general concept. Successors and popularizators of Appolonius would have IMO arrived at the general use of coordinates, but apparently Apollonius was the endpoint of antique geometry. Even his treatise looks unfinished. – Thomas Riepe Mar 22 '10 at 8:57
A further point may be that ancient geometry teaching worked by personal instruction, books were probably guides for the teachers and not for the students. This would have blocked the normal way of publishing simplyfied popularizations, where a general and systematical use of coordinates could have emerged. – Thomas Riepe Mar 22 '10 at 8:58
up vote 19 down vote accepted

Let V be the vertex of a parabola, F its focus, X a point on its symmetry axis, and A a point on the parabola such that AX is orthogonal to VX. It was well within the power of the Greeks to prove relations such as $VX:XA = XA:4VF$. If you introduce coordinate axes, set $x = VX$, $y = XA$ and $p = VF$, you get $y^2 = 4px$, the modern form of the equation of a parabola.

Everything now depends on what "invention of coordinate geometry" means to you. I do not think that the Greeks' work on conics should be confused with coordinate geometry since they did not regard the lengths occurring above as coordinates. It's just that parts of their results are very easily translated into modern language.

In a similar vein, Eudoxos and Archimedes already were close to modern ideas behind integration, but they did not invent calculus. Euclid, despite Heath's claim to the contrary, did not state and prove unique factorization. And Euler, although he knew the product formula for sums of four squares, did not invent quaternions (Blaschke once claimed he did). In any case, we are much more careful now with sweeping claims such as "Appolonius knew coordinate geometry" than historians were, say, 100 years ago.

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+1 for putting across what I was trying to say in my earlier comment, but doing so much better than I was able to – Yemon Choi Mar 22 '10 at 19:35

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